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I'm struggling with the following problem:

Let $\{M_i\}_{i\in I}$ be a directed system of $R$-modules and let $J\subset I$ be a cofinal directed subset. Let $\{M_j\}$ be the directed system induced by the system over $I$. Prove that the direct limits over $I$ and over $J$ are isomorphic to each other as $R$-modules.

My attempt: I already constructed the direct limit over $I$ as $M/N$, where $M=\oplus_{i\in I}M_i$ and $N=\langle x_i - f_{ij}(x_i): x_i \in M_i,\: i,j\in I,\:i \preceq j \rangle$.(Here I abused the notation, i.e. $x_i$ is actually $i_i (x_i)$, where $i_i :M_i \to M$ is the canonical embedding.)

Similarly, I constructed the direct limit over $J$ as $M_J / N_J$, where $M=\oplus_{j\in J}M_j$ and $N_J=\langle x_i - f_{ij}(x_i): x_i \in M_i,\: i,j\in J,\:i \preceq j \rangle$.

Now I tried to construct an isomorphism $\phi: M_J/N_J\to M/ N$ as $\phi(a+N_J)=a+N$. $\phi$ is clearly a well-defined $R$-module homomorphism. Also, by cofinality, one can show easily that $\phi$ is onto.

However, I stuck when I tried to prove that $\phi$ is one-to-one. i.e. its kernel is trivial. If $\phi(a+N_J)=N$, then it implies that $a= (a_i)_{i\in I}\in N$. Here $a_i=0$ if $i \notin J$. How should I prove that $a \in N_J$ here?

Any hints or advices will help a lot!

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  • $\begingroup$ Do you know the the definition of direct limit of universal property?maybe you can find this in Wikipedia $\endgroup$ – Sky May 18 '18 at 5:03

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